Verify if the function f(x) = xarctanx - ln(1+x^2) is concave upwards .
If a function is concave upwards then it's second derivative is positive.
We'll determine the 1st derivative:
f'(x) = arctanx + 1/(1+x^2) - 2x/(1+x^2)
f'(x) = arctanx – x/(1+x^2)
We'll determine the 2nd derivative
f"(x)= 1/(1+x^2) – (1+x^2-2x^2)/(1+x^2)^2
f"(x) = (1+x^2+x^2-1)/(1+x^2)^2
We'll combine like terms inside brackets:
f"(x)=2x^2/(1+x^2) >= 0
Since the second derivative is always positive , for any real value of x, then the function is concave upwards.